Besides the two general techniques mentioned above for hardness of approximation in P, and their applications, there are also results that follow from problem-specific tweaks to the exact-lower-bound constructions.
In the context of distance computations in graphs, some examples are for APSP [1]}, [2]}, Diameter and related problems (without the use of graph-products) [3]}, [4]}, [5]}, for the Girth in directed graphs [6]} (see [7]}, [8]} for recent upper bounds for this problem), and for dynamic near-additive spanners [9]}.
A more general result talks about the possibility of avoiding the \(\log {W}\)  factor that comes from the standard scaling trick in \((1+\varepsilon )\) -approximations for a problem with weights up to \(W\)  by reduction to an unweighted problem [10]}.
Moreover, there is a connection between deterministic approximation algorithms and circuit complexity that has lead to strong inapproximability results [11]}, [12]}, [13]} but the hardness assumptions underlying such barriers are known to be breakable with randomized algorithms.
